Spherical Trigonometry — Mathematics Knowledge System

Overview

Spherical trigonometry studies triangles drawn on the surface of a sphere, whose sides are arcs of great circles (the shortest paths between two points on a sphere) rather than straight line segments. Because the surface is curved, the familiar plane rules — angles summing to 180°, similar triangles, the ordinary law of sines and law of cosines — no longer hold exactly; spherical triangles have angle sums strictly greater than 180°, and their own analogues of the sine and cosine rules relate sides (measured as angles subtended at the sphere's center) to the triangle's vertex angles. This is not an abstract curiosity: it is the mathematics behind navigation, astronomy, and geodesy, since the Earth is (to a good approximation) a sphere and the shortest flight or ship route between two distant cities follows a great-circle arc.

Intuition

On a flat plane, the law of cosines c² = a² + b² − 2ab cos C relates a triangle's third side to the other two sides and the included angle. On a sphere, sides are no longer lengths but angles (the angle each arc subtends at the sphere's center, since arc length = radius × angle and the radius is fixed), so the spherical law of cosines replaces the algebraic combination a² + b² − 2ab cosC with the trigonometric combination cos a cos b + sin a sin b cos C — a formula that reduces to the plane law of cosines when the sides are small compared to the sphere's radius (a regime where cos x ≈ 1 − x²/2 and sin x ≈ x recover the familiar flat formula). The extra angle sum beyond 180° — the spherical excess — is a direct measure of how much the triangle 'bulges' over the curved surface, and by Girard's theorem it is proportional to the triangle's area.

Formal Definition

Definition

On a sphere of radius 1, a spherical triangle has sides a, b, c (arc lengths, measured as the angle each subtends at the center) opposite vertex angles A, B, C:

\cos c = \cos a\cos b + \sin a \sin b\cos C

Spherical law of cosines (for sides)

\frac{\sin A}{\sin a} = \frac{\sin B}{\sin b} = \frac{\sin C}{\sin c}

Spherical law of sines

A + B + C > 180^\circ

Spherical excess: angle sum exceeds 180° (the excess is proportional to the triangle's area)

Worked Examples

Apply the spherical law of cosines.
$\cos c = \cos60^\circ\cos90^\circ + \sin60^\circ\sin90^\circ\cos45^\circ$
Substitute known values: cos60°=0.5, cos90°=0, sin60°≈0.8660, sin90°=1, cos45°≈0.7071.
$\cos c = (0.5)(0) + (0.8660)(1)(0.7071) \approx 0.6124$
Take the inverse cosine.
$c = \arccos(0.6124) \approx 52.24^\circ$

Answer: c ≈ 52.24°

Practice Problems

Difficulty 6/10

A spherical triangle has sides a = 90°, b = 90°, and included angle C = 90°. Find side c using the spherical law of cosines.

Difficulty 7/10

In a spherical triangle, sin A/sin a = sin B/sin b. If a = 60°, b = 90°, and A = 30°, find sin B and angle B.

Difficulty 8/10

An aircraft navigator needs the great-circle angular distance between two waypoints at latitudes 40°N and 50°N whose longitudes differ by 30°. Using the navigational form of the spherical law of cosines, find the angular distance and the great-circle distance in kilometers (Earth radius ≈ 6371 km).

Quiz

Summary

Spherical triangles are formed by great-circle arcs on a sphere; their sides are measured as central angles, not lengths.
The spherical law of cosines, cos c = cos a cos b + sin a sin b cos C, is the curved-surface analogue of the plane law of cosines.
The spherical law of sines, sinA/sina = sinB/sinb = sinC/sinc, parallels the plane law of sines.
Angle sums exceed 180° (spherical excess), proportional to the triangle's area — a direct signature of the sphere's curvature.
This machinery underlies navigation and geodesy: great-circle distances between points on Earth are computed with the spherical law of cosines.

References

WebsiteWikipedia — Spherical trigonometry
WebsiteWikipedia — Great-circle distance